Wave filter



H. W. BODE March 24, 1936;

WAVE FILTER Filed March 24, 1933 2 Sheets-Sheet l FREQUENCY I l k FREQUENCY l0 FREQUENCY INVENTOR H. M. BODE "BY/W ATTORNEY March 24, 1936. w. BQDE 2,035,258

WAVE FILTER Filed March 24, 1955 2 Sheets-Sheet 2 F76. FIG. /2 179/3 INVENTOR H W 5005 ATTORNEY Patented Mar. 24, 1936 WAVE FILTER Hendrik W. Bode, New York, N. 1Y., assignor to Bell Telephone Laboratories, Incorporated,. New :York,.N. Y., a corporation :of'New'Y'ork Application March 24, 1933, Serial No. 662,412

9 Claims.

This invention relates to broad-band wave filters and has for its principal objects to improve the selectivity and to extend the'range of transmission characteristics of such filters.

In accordance with the invention,broad-band wave filters are provided having peaks of infinite attenuation at frequencies close to or at the cutofi frequencies of .the transmission band,.or within the band itself. By an alternative'form of the invention attenuation characteristics having a finite maximum ofattenuation at any desired frequency outside the band are provided, this maximum being independent of the presence or absence of dissipation in the branches of the network.

The nature of theinventi'on will be more clearly understood from the following detailed description and by reference to the appended drawings of which:

Figures 1, 2, Band 4 'are'schematics illustrative of the principles of the invention;

Fig. 5v shows one embodiment of the invention;

Fig. 6 is the prototype network corresponding .to Fig. 5;

Figs. 7, 8, and 9 illustrate .characteristicsof the networks of Fig. '5; V

Fig. 10 is a modified form of the network 'of Fig. 5;

Figs. 11 to 19 inclusive, '"show additional forms of the networks of the invention together with their prototypes;

Figs. 20 and 21 'are explanatory of a theorem used in connection with further modifications of the invention, and

Fig. 22 illustrates a furthermodification based on this theorem.

One method of introducing attenuation peaks into the transmission characteristic of a wave filter is described in U. S. Patent 1,538,964, issued May 26, 1925 to O. J. Zobel, which shows the so-' called m-type filter and explains the manner of deriving it from elementary ladder type "filters having no attenuation peaks. While it is possible to design filters of this type in which the peaks are located as close as may be desired to the cut-off frequencies, it has been found in practice that filters constructed in accordance with such designs do not exhibit the expected properties. The reason for this is that the normal dissipation in the filter elements, which in any case prevents the attenuation from rising to an infinite value at the peaks, becomes increasingly effective as the peaksare moved closer to the cutoff frequencies, tending to limit the attenuation to smallerand smaller values,

multiply the line brance impedances by the factor m and' to divide the lattice branches byzthexsame factor. This is illustrated by the network of Fig.

2. The image impedance :K1 and the transfer constant 01 of the network of Fig. 1 are given by K.=\ .z.

and

and the corresponding quantities K2 and 02 for Fig. .2 aregivenby The process of 'm-derivation has not changed tanh and

the image impedance but has-modifiedthe transfer constant by theintroduction of the factor m as a multiplier of the hyperbolic tangent of the half transfer constant.

When the factor m is positive and less than unity, the network of Fig. 2 can betransformed into a symmetrical T with physically realizable impedances, corresponding to the ladder networks of theZobel patent. When m is greater than unity, this transformation can not be ac- 'complished, except in a few special cases, al-

though the m-derived latticefis evidently physically realizable.

If two m-derived lattices having thesameimage impedances but different values of .m, namely mirand 'mz are connec'teddn tandem as illustrated by Fig. 3, it may bez'shown that the combination is equivalent to the network of Fig. 4 which is also a symmetrical latticehaving line branch impedances Za. made up of impedances (m1+m2) Z1 and zirmiamn/mimz in tparalleL 'andfllattlce 1 1+tanh 2 tanh i If now, the transfer constant of the Z1Zz p1Ot0- type of the component lattices of Fig. 3 be designated by 60, it follows that 01 0 7 V tanh -m tanh 2 and 1 tanh- =m tanh and, hence, that (m -i-m tanh '1 pa m; tanh It is readily seen by comparing the products of the line and lattice impedances in each case, that each lattice of Fig. 3 and that of Fig. 4 has the same image impedance, namely,

multiplying both sides of Equation 4 by K gives 6 1 0 1+n n m K tanh a? tanh 5 From the general relationship between the image impedance and the transfer constant the quantity 'K tanh (5) is identified as the line branch impedance of Fig. 3 and, since likewise,

Equation 5 becomes 1+ 2 )z z l+m2) i( mlmz 2 Z: Z2+m1mzZ1 )Z 1+ 2 .7 1 2 1 2 Equation 6 is readily identified as the parallel combination of the two impedances constituting the line branches in Fig. 4. The value of Zb may be developed in the same way, using the relationship In the general networkof Fig. 4 the ratio of each'parallel connected impedance in the line branches to the corresponding series impedance component in the lattice branches has the value which for real valuesof m has a minimum posthe impedance multiplying factors in the network of Fig. 4 have the values 1111+ m 2a and which are positive real quantities so long as a is positive. The impedance ratio referred to above has the value 1+ 2) 4&2

11111212 & +b

which obviously, must be less than 4.

.With this relationship in mind specific networks of the invention having the general schematic form of Fig. 4 are readily arrived at.

One example of the networks of the invention is shown in Fig. 5, the case illustrated being that of a low-pass filter. The simple filter structure from which this network may be derived in the above indicated manner is the symmetrical lattice shown in Fig. 6 which has line branches consisting of inductances L and lattice branches consisting of simple resonant combinations of inductance L in series'with capacities C.

In the network of Fig. 5 each line branch consists of a parallel combination of an inductance L01 with a resonant circuit L101 and each lattice branch consists of a simple series combination of an inductance L2 and capacity C2. The'values of the elements when the m-factors have the complex values of Equation 7 are as follows:

' Representative impedance and attenuation characteristics for this network are shown in Fig. '7 for the case in'which the coeflicients a and b have the values 0.36 and 0.217 respectively. Curve III shows the variation with frequency of the line branch reactance and curve I l the variation of the lattice branch reactance. The band cut-off occurs at the frequency fc at which the combination L101 is resonant. The lattice branch L202 resonates at a lowerfrequency ii at which the line branch is anti-resonant.

It will be noticed that in the range above the cut-off frequencies the two reactances become most nearly equal at a frequency f2 not far removed from the cut-off. At this frequency the lattice, regarded as a bridge, is most nearly balanced and the current transmitted through it is therefore a minimum. This is illustrated by curve IZof ;the;figure which shows, to an arbitrary scale, the frequency variation of the attenuation constant of the network. Instead of rising to a sharp peak, as in the m-derived filters of Zobel Patent 1,538,964, the attenuation increases to a finite maximum determined by the ratio of the reactances.

The location of the frequency of maximum attenuation with respect to the cut-off depends on the value of the quantity a +b If this quantity is very small relatively to unity the maximum will occur very close to the cut-off frequency and will move away from the cut-off frequency as the value is increased towards unity. For values greater than unity the attenuation maximum is absent from the characteristic. The height of the attenuation maximum may be increased as much as desired by making the coefiicient b small with respect to a.

The characteristics discussed above are those which would be obtained with dissipationless impedances. If the effect of dissipation is taken into account, it is possible, by proportioning the complex ms in accordance with the dissipation to obtain additional characteristics of unique types. This will be illustrated by the consideration of how the network of Fig. may be designed to have an infinite attenuation peak at its cutoff frequency,

If the coils in the lattice of Fig. 6 have the same dissipation factor, defined as the ratio of reactance to resistance, then the line and the lattice impedances, denoted by Z1 and Z2 respectively, may be expressed as it-sea] where Q is the dissipation factor, to denotes 21r times frequency, and we corresponds to the cut-off frequency.

The transfer constant 0 is given by which, at the cut-off frequency, becomes and If a network m-derived from Fig. 6 could be constructed with its m factor of the value 1 w/ +jQ The transfer constant, 0m, of that network would have the value defined by 0 0 tanh tanh 5 and tions of .the-two components are of opposite .sign

at most frequencies, that is, the conjugate comcomplex quantity 1+y'Q is given by from which the values of the conjugate ms may be determined as 1 The impedance coefficients have the following values and m +m 2a 1 1 m"a +b E the value of Q being, of course, that corresponding to the cut-off frequency.

If the value of Q be assumed equal .to 50, which is representative of air core inductances of commercial type, the various factors have the following numerical values:

The substitution of these numerical values in Equations 9 gives the values of the elements of Fig. 5 in terms of the L and C of the prototype for the case under consideration. These are:

L1= 10.1L L2: 5.051; (.18) C1: C2: 202C From a physical standpoint, the manner in which the peak at the cut-off is produced may best be appreciated by a consideration of the frequency variation of the line and. lattice impedances. This is illustrated by the curves of Fig. 8 in which curve l3 represents the variation of the line branch reactance, curve I4 the line branch effective resistance and curve [5 and straight line l6 represent respectively the reactance and resistance of the lattice branch. For the sake of clarity, the resistance curves are drawn to a greater ordinate scale than the reactance curves.

It is to be observed that anti-resonance and resonance effects in the line branch, which in the absence of dissipation would make curve'l3 similar to curve H! of Fig. 9, appear only as'a slight irregularity at the frequency f1. Thisis due to the fact that the inductance L01 is relatively small and the shunting impedance 'L1C1 is a'resonant circuit of great stiffness, the combined reactanoe being substantially that of inductance L61. A second result of the proportions of this combination is that the efiective resistance at the anti-resonance frequency is quite small, its value being approximately equal to the resistance of the shunting circuit L1C1. At the cut-off frequency the resistance falls to half this value and is equal to the resistance of the lattice branch. For the case under consideration, the balance of the two impedances at the cut-off may be described broadly as follows. The reactance of the lattice arm is balanced by the low reactance of the'shunting inductance L01 in the line branch while the resistance of the lattice arm is balanced substantially by the resistance of the resonant circuit L101 in the line branch, this resistance, by virtue of the parallel combination in the line branch appearing only in the neighborhood of the desired frequency.

The attenuation characteristic of the computed network is illustrated by curve ll of Fig. 9 which shows the attenuation in decibels plotted as a function of frequency, the curve being approximately to scale. The cut-off is extremely sharp and above the suppression peak there is a reasonably high attenuation which can be supplemented if desired by additional filter sectors of the ordinary type.

It is evident from Equation 11 that the quantity is complex at all frequencies and, hence that, by suitably choosing the complex m factors, the suppression peak may be located anywhere in the frequency range including the transmission band. The procedure in computing the mfactors will be the same as outlinedabove, the value of Q in each case being that corresponding to the frequency for which suppression is desired. The form of the resulting network, of course, remains unchanged.

The attenuation characteristic of a network like that of Fig. 5, but having a suppression peak in the middle of the band is illustrated by curves l8 of Fig. 9. For this case the value of Q was again assumed to be 50, giving for the values of the m factor m1=.022+7'1.733 and It will be noticed that the suppression peak is extremely sharp and falls'very quickly to zero at frequencies quite close to the peak. The attenuation above the cut-off is also very small.

Numerical values of the actual inductances and capacities for the two cases illustrated are given in the table below, assuminga cut-off frequency of 3000 cycles per second and a characteristic impedance at zero frequency of 600 ohms.

Value Element g? Peak at mid band 1.54 millihenries 51 millihenries 2630 .nillihenries 5. 48 microfarads .00425, microfarads 50 The general form of the networks of the invention, as illustrated by Fig. 4, is that of a symmetrical lattice and, therefore, best adapted for use formationof the symmetrical lattice and these unbalanced equivalents can be used in circuits having one side grounded. Usually the possibility of this transformation can be ascertained by inspection of the developed lattice network, the requirement being that the lattice arm should contain an impedance in series corresponding to one of the parallel connected impedances in the line arm but of greater magnitude. In the suppress'ion networks of the invention, the values of the m factors are such that this is practically always the case.

The bridged-T equivalent of the network of Fig. 5 is shown in Fig. 10, the element values being designated in terms of those of Fig. 5.

The prototype lattice, Fig. 6, from which the networks of Figs. 5 and are derived is equivalent to a simple T-network having series inductances L and a shunt capacity 2C, that is, to a mid-series terminated low-pass filter section of the so-called constant-k type. It is obvious that other forms of the networks of the invention may be arrived at using other simple networks as prototypes. Two additional forms are illustrated by the lattice of Fig. 11 and the bridged-T of Fig. 12, both of which are derived in the manner described above from the prototype lattice of Fig. 13 which corresponds to a ladder type mid-shunt terminated low-pass filter section,

having a series inductance 2L and shunt capacities C at each end.

Further examples of the networks of the in,- vention are shown in Figs. 14 and 15 which are derived from the mid-shunt high-pass filter of Fig. 16 and by Figs. 1'7 and 18 which are bandpass structures derived from the elementary band-pass filter of Fig. 19. The inductance and capacity values of the complex networks may be determined directly from the designated coefilcients in the prototypes by the application of the relationships indicated by Figs. 3 and 4, and Equations 8. In each case the complex network has the same image impedance as the prototype, but has a transfer constant modified in accordance with the values of the complex ms. The procedure in the design of these networks to produce peaks of infinite attenuation at any desired frequencies is the same as in the case of the example already discussed. The steps comprise, first, developing an expression for the hyperbolic tangent of the half transfer constant for the dissipative case, and, second, determining the particular complex value of this quantity at the chosen frequency, the inverse of which gives the first of the required conjugate complex m-factors.

It will be noticed from the numerical values given for the low-pass filter case that the impedance elements may have greatly disparate Values. For example, in the case where the peak is made to occur at half the cut-off frequency, the largest inductance in the network is about five thousand times as great as the smallest. While this does not render the design impracticable, it is frequently desirable for manufacturing purposes to have the element values more nearly of the same order of value, as this permits the same type of coil construction to be used throughout. This is accomplished in a modified form of the invention in which the network is constructed as the equivalent of the combination of the prototype network together with the two complex-m derivatives thereof. The principle of the combination is illustrated by Figs. 20 and 21 of which Fig. 21 shows the single symmetrical lattice which is the equivalent of the tandem connection of the two lattices of Fig. 20 when these have the same image impedances.

The two lattices of Fig. 20 have line impedances X1 and X2 respectively and lattice impedances Y1 and Y2 respectively, these being related so that X1Y1=X2Y2 to make the image impedances equal. The equivalent lattice of Fig. 21 has for its line branches the parallel combination of (X1+X2) with (Y1+Y2) and for its lattice branches the series combination of these impedances in parallel. Using this relationship, it is possible to construct a single lattice equivalent to the combination of the general network of Fig. 4 with its prototype Fig. 1, or, as a specific example, the combination of the complex network of Fig. 5 with its prototype Fig. 6.

The branches of the network obtained by the direct application of this equivalence relationship to Figs. 5 and 6 will be of a rather complex character but by the application of the equivalent transformations described by O. J. Zobel in an article on Theory and design of uniform and composite electric wave filters, Bell System Technical Journal Vol. II, No. 1, January, 1923, pages 45 and 46, the network may be reduced to the relatively simple form shown in Fig. 22. In terms of the a and b coeificients of the complex m-factors as given by Equations 7, the elements of the network of Fig. 22 have the following values:

In this network the disparity of the element values is greatly reduced, for example, in the case of the low-pass filter having a peak at the mid-band frequency the ratio of the largest to the smallest inductance is reduced from about 5000 to about 164. The impedance transformations referred to may, evidently, be applied in various ways giving other forms of the final network and in certain cases one or others of these possible modifications may give better uniformity of the element values. The best form in any particular case may usually be arrived at by trial.

The transfer constant of the combination network of Fig. 22 will, of course, be the sum of the transfer constants of the component sections and the inclusion of the prototype section will have the effect of increasing the attenuation at frequencies beyond the cut-off. For the case in which the suppression peak is located at the cutoff the attenuation above the cut-off is shown by curve IQ of Fig. 9.

What is claimed is:

1. A symmetrical four-terminal transmission network comprising a path between one input terminal and one output terminal, a plurality of reactance elementsincluded in said path con-.- stituting an impedance characterized by at least one resonance and one anti-resonance at diifere ent frequencies, av second path between the same input terminal and the other output terminaLa plurality of reactance elements included in said second path constituting an impedance having a resonance at substantially the same frequency as the frequency of the said anti-resonanceof the impedance in the first mentioned path, the reactances of the elements of said impedances being proportioned with respect to each other to providea transmission band between preassigned frequencies and being further proportioned with respect to the dissipation factors of the elements whereby the effective resistance and the. reactance of the first path are simultaneously balanced by the effective resistance and the reactance, respectively, of the said second path at a frequency close to the frequency of the said anti-resonance of the first path and whereby a peak of substantially infinite attenuation is provided atthe frequency of balance.

2. A network in accordance with claim 1 in which the .said two paths constitute respectively a line branch and a lattice branch of a symmetrical lattice network.

3. A network in accordance with claim 1 in which the said path including parallel connected impedances constitutes a line path between an input and an output terminal and in which the other path includes a portion connected from the mid-point of one of the parallel connected impedances to the other input and output terminals in common.

4.. A network in accordance with claim 1. in which the impedances are proportioned to provide substantially infinite attenuation at one of the cut off frequencies of the transmission band determined by the reactances of the elements constituting the network branches.

5. A network in accordance with claim 1 in which the impedances are proportioned to provide substantially infinite attenuation at a fre-' quency within the transmission band determined by the reactances of the elements constituting the network branches.

6. A wave transmission network comprising two pairs of impedance branches connected to form a symmetrical lattice between a pair of input terminals and a pair of output terminals, the line branches of said lattice each comprising two parallel connected reactive impedances of different frequency characteristics and the lattice branches each comprising two series connected impedances similar respectively to the impedances of the line branches, the reactances of said impedances being proportioned with respect to each other to provide 'a transmission band between preassigned frequencies, and the parallel connected reactances of the line branches being related in magnitude to the respectively, similar series connected reactances of the lattice branches by a numerical factor less than four.

7. A symmetrical four-terminal transmission network comprising a path between one input terminal and one output terminal including an anti-resonant circuit of low impedance at frequencies other than that of anti-resonance, a second path between the same input terminal and the other output terminal including a series resonant circuit of high impedance at frequencies other than that of resonance, the resonance frequency of said second circuit being the same as the anti-resonance frequency of said first circuit, and the reactances and the effective resistances due to dissipation in the reactance elements being proportioned to balance each other at. a frequency substantiallyequal to the said anti-reasonance frequency whereby the transmission of currents of this frequency is suppressed.

8. A symmetrical four-terminal transmission network comprising a path between one input terminal and one output terminal including two reactive impedances of difierent frequency characteristics connected in parallel, the parallel combination of said impedances having an antiresonance at a preassigned frequency, and a second path between the same input terminal and the other output terminal including a reactive impedance having a resonance at the said anti-resonance frequency of the parallel connected impedances and being proportioned with respect to said parallel combination to provide a broad transmission band, said parallel connected impedances being also proportioned so that their joint reactance and their effective resistance due'to dissipation in the reactance elements balance the reactance and effective resistance of the said second path at a frequency substantially equal to the said anti-resonance frequency, whereby the transmission of currents of this frequency is suppressed.

9. A lattice type broad band Wave filter comprising two equal line branches of impedance Z9. and two equal 1attice branches of impedance Zn and having the same characteristic impedance as that of a prototype lattice filter having line branches of impedance Z1 and lattice branches of impedance Z2, the impedances Z1 and Z2 being constituted'by reactance elements proportioned to provide a transmission band between preassigned frequencies, the impedances Za. being equal in value to the impedance of the parallel connection of (m1+m2) Z1 and and the impedances Zb being equal in value to the series connection of impedances Z1 and Z2 HENDRIK W. BODE. 

